Fill a large steamer or pot with about 2 inches of water. Squeeze the juice of 1 lemon into the water, then add the lemon halves and 1 tablespoon sea salt to the pot. Bring to a boil over medium heat.
Place the lobsters in the steamer basket or directly in the pot; cover and steam until the shells turn bright red, 8 to 10 minutes. Remove from the pot with tongs and rinse under cold water to cool slightly.
Remove the meat from the lobster shells: Twist off the claws, then break off the tail. Pull off the flippers. Insert your thumb into the flipper end of the tail and force out the meat. Crack the claws with the flat side of a knife or a lobster cracker; remove the meat. Roughly chop the lobster meat. Transfer to a bowl and chill 15 minutes.
Meanwhile, whisk the mayonnaise, 4 tablespoons melted butter and the juice of the remaining lemon in a large bowl. Add the lobster meat and toss to coat. Season with salt.
Heat a grill pan or skillet over medium heat. Brush the outside of the hot dog buns with 2 tablespoons melted butter. Toast the buns until slightly golden, about 1 minute per side. Divide the lobster salad among the buns. Drizzle with the remaining 1 tablespoon melted butter and sprinkle with the celery leaves and celery salt.